Solving Polynomial Equations Research Articles

New Theorems Solving Fifth Degree Polynomial Equation in Complete Forms by Proposing New Five Roots Composed of Radical Expressions

This paper is presenting new theorems and formulas to solve fifth degree polynomial equation in general forms by proposing five formulary solutions that we can calculate nearly simultaneously. The proposed roots for fifth degree polynomial in this paper are composed of radical expressions distributed in a specific architecture that allows them to neutralize each other during multiplication by eliminating radicality and reducing degrees of terms. The proposed architecture for each solution of quantic polynomial is composed of at least six distributed terms, where each term is a multiplication of a pair of radical expressions, whereas multiplications among proposed terms is allowing the reduction of degrees in order to obtain a quartic form at maximum. Furthermore, the proposed architectures of radical roots in this paper are allowing also to solve polynomial equations with degrees higher than five by reducing their expressions. As a result, this paper is presenting two new theorems to solve quantic equation in general forms, where one theorem is eliminating the expression of fourth degree then reducing the whole fifth degree equation into a simplified form, whereas the other new theorem is solving the quantic equation without eliminating the expression of fourth degree. All proposed theorems in this paper are built on a scaling logic concretized by engineering the structure of solutions then calculating the precise formulas of roots for equations, which allow to calculate all roots nearly simultaneously and forward the used engineering methodology to solve polynomial equations with degrees higher than five.

Compact Circuits for Efficient Möbius Transform

The Möbius transform is a linear circuit used to compute the evaluations of a Boolean function over all points on its input domain. The operation is very useful in finding the solution of a system of polynomial equations over GF(2) for obvious reasons. However the operation, although linear, needs exponential number of logic operations (around n · 2n−1 bit xors) for an n-variable Boolean function. As such, the only known hardware circuit to efficiently compute the Möbius Transform requires silicon area that is exponential in n. For Boolean functions whose algebraic degree is bound by some parameter d, recursive definitions of the Möbius Transform exist that requires only O(nd+1) space in software. However converting the mathematical definition of this space-efficient algorithm into a hardware architecture is a non-trivial task, primarily because the recursion calls notionally lead to a depth-first search in a transition graph that requires context switches at each recursion call for which straightforward mapping to hardware is difficult. In this paper we look to overcome these very challenges in an engineering sense. We propose a space efficient sequential hardware circuit for the Möbius Transform that requires only polynomial circuit area (i.e. O(nd+1)) provided the algebraic degree of the Boolean function is limited to d. We show how this circuit can be used as a component to efficiently solve polynomial equations of degree at most d by using fast exhaustive search. We propose three different circuit architectures for this, each of which uses the Möbius Transform circuit as a core component. We show that asymptotically, all the solutions of a system of m polynomials in n unknowns and algebraic degree d over GF(2) can be found using a circuit of silicon area proportional to m · nd+1 and circuit depth proportional to 2 · log2(n − d).In the second part of the paper we introduce a fourth hardware solver that additionally aims to achieve energy efficiency. The main idea is to reduce the solution space to a small enough value by parallel application of Möbius Transform circuits over the first few equations of the system. This is done so that one can check individually whether the vectors of this reduced solution space satisfy each of the remaining equations of the system using lower power consumption. The new circuit has area also bound by m · nd+1 and has circuit depth proportional to d · log2 n. We also show that further optimizations with respect to energy consumption may be obtained by using depth-bound Möbius circuits that exponentially decrease run time at the cost of additional logic area and depth.

On a Fifth-Order Method for Multiple Roots of Nonlinear Equations

There are several measures for comparing methods for solving a single nonlinear equation. The first is the order of convergence, then the cost to achieve such rate. This cost is measured by counting the number of functions (and derivatives) evaluated at each step. After that, efficiency is defined as a function of the order of convergence and cost. Lately, the idea of basin of attraction is used. This shows how far one can start and still converge to the root. It also shows the symmetry/asymmetry of the method. It was shown that even methods that show symmetry when solving polynomial equations are not so when solving nonpolynomial ones. We will see here that the Euler–Cauchy method (a member of the Laguerre family of methods for multiple roots) is best in the sense that the boundaries of the basins have no lobes. The symmetry in solving a polynomial equation having two roots at ±1 with any multiplicity is obvious. In fact, the Euler–Cauchy method converges very fast in this case. We compare one member of a family of fifth-order methods for multiple roots with several well-known lower-order and efficient methods. We will show using a basin of attraction that the fifth-order method cannot compete with most of those lower-order methods.

Exact Expressions for Trigonometric Functions

Summary Solving polynomial equations and trigonometry are typically taught in separate modules or even separate classes. The derivation of a one-third angle formula gives students the opportunity to tie these two topics together. An algorithm for finding the cosine of one-third of a prescribed angle is based on Cardano’s formula. In addition, we show how this algorithm can be combined with half, double, and triple angle formulas to give exact expressions for trigonometric functions written in terms of radicals for an infinite number of angles between 0 and π / 2 .

Matrix approach to solve polynomial equations

Polynomials are widely employed to represent numbers derived from mathematical operations in nearly all areas of mathematics. The ability to factor polynomials entirely into linear components allows for a wide range of problem simplifications. This paper presents and demonstrates a novel straightforward approach to solving polynomial problems by converting them to matrix equations. Each polynomial of degree n can be decomposed into a sum of degree ⌈n2⌉ polynomials squared, i.e., ∑i=0naixi=∑i=1⌈n2⌉+1∑j=0j=⌈n2⌉bi,jxj2. It follows that the complexity of factorizing a polynomial of degree 2n is equivalent to that of factorizing polynomial of degree 2n−1. The proposed method for solving fourth-degree polynomials will be a valuable contribution to linear algebra due to its simplicity compared to the current method. This work presents a unique approach to solving polynomials of four or fewer degrees and presents new possibilities for tackling larger degrees. Additionally, our methodology can also be used for educational purposes.

On family of the Caputo-type fractional numerical scheme for solving polynomial equations

Fractional calculus can be used to fully describe numerous real-world situations in a wide range of scientific disciplines, including natural science, social science, electrical, chemical, and mechanical engineering, economics, statistics, weather forecasting, and particularly biomedical engineering. Different types of derivatives can be useful when solving various fractional calculus problems. In this study, we suggested a single step modified one parameter family of the Caputo-type fractional iterative method. Convergence analysis shows that the proposed family of methods' order of convergence is . To determine the error equation of the proposed technique, the computer algebra system CAS-Maple is employed. To illustrate the accuracy, validity, and usefulness of the proposed technique, we consider a few real-world applications from the fields of civil and chemical engineering. In terms of residual error, computational time, computational order of convergence, efficiency, and absolute error, the test examples' acquired numerical results demonstrate that the newly proposed algorithm performs better than the other classical fractional iterative scheme already existing in the literature. Using the computer program Mathematia 9.0, we compare the draw basins of attraction of the suggested fractional numerical algorithm to those of the currently used fractional iterative methods for the graphical analysis. The graphical results show how quickly the newly developed fractional method converges, confirming its supremacy to other techniques.

Polynomial identities satisfied by generalized polynomials

The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in \mathbb{F}[x]$ and $Q\in \mathbb{C}[x]$ be polynomials. Our aim is to prove characterization theorems for generalized polynomials $f\colon \mathbb{F}\to \mathbb{C}$ of degree two that also fulfill equation \[ f(P(x))= Q(f(x)) \] for each $x\in \mathbb{F}$. As it turns out, the difficulty of such problems heavily depends on that we consider the above equation for generalized polynomials or for (normal) polynomials. Therefore, firstly we study the connection between these two notions.

A novel data-driven approach on inferring loop invariants for C programs

Inspired by the procedure that experts construct loop invariants, we propose a novel data-driven approach to automatically infer loop invariants. The approach consists of two phases, data-driven inference of candidate invariants and validity check of loop invariants. Inspired by the procedure that experts construct loop invariants, we propose a novel data-driven approach to automatically infer loop invariants for C programs. The approach consists of two phases, data-driven inference of candidate invariants and validity check of loop invariants. The first phase generates candidate invariants by solving polynomial equations and synthesizing the extended loop conditions. The second phase prunes out spurious predicates and redundant predicates in the candidate invariants. The experimental results demonstrate that the proposed approach generates valid invariants for 35 benchmarks out of 38. The proposed approach costs less time to generate more informative and precise invariants than the state-of-the-art methods.

Constructing Dixon Matrix for Sparse Polynomial Equations Based on Hybrid and Heuristics Scheme

Solving polynomial equations inevitably faces many severe challenges, such as easily occupying storage space and demanding prohibitively expensive computation resources. There has been considerable interest in exploiting the sparsity to improve computation efficiency, since asymmetry phenomena are prevalent in scientific and engineering fields, especially as most of the systems in real applications have sparse representations. In this paper, we propose an efficient parallel hybrid algorithm for constructing a Dixon matrix. This approach takes advantage of the asymmetry (i.e., sparsity) in variables of the system and introduces a heuristics strategy. Our method supports parallel computation and has been implemented on a multi-core system. Through time-complexity analysis and extensive benchmarks, we show our new algorithm has significantly reduced computation and memory overhead. In addition, performance evaluation via the Fermat–Torricelli point problem demonstrates its effectiveness in combinatorial geometry optimizations.

Extending the Local Convergence of a Seventh Convergence Order Method without Derivatives

For the purpose of obtaining solutions to Banach-space-valued nonlinear models, we offer a new extended analysis of the local convergence result for a seventh-order iterative approach without derivatives. Existing studies have used assumptions up to the eighth derivative to demonstrate its convergence. However, in our convergence theory, we only use the first derivative. Thus, in contrast to previously derived results, we obtain conclusions on calculable error estimates, convergence radius, and uniqueness region for the solution. As a result, we are able to broaden the utility of this efficient method. In addition, the convergence regions of this scheme for solving polynomial equations with complex coefficients are illustrated using the attraction basin approach. This study is concluded with the validation of our convergence result on application problems.

Efficient product formulas for commutators and applications to quantum simulation

We construct product formulas for exponentials of commutators and explore their applications. First, we directly construct a third-order product formula with six exponentials by solving polynomial equations obtained using the operator differential method. We then derive higher-order product formulas recursively from the third-order formula. We improve over previous recursive constructions, reducing the number of gates required to achieve the same accuracy. In addition, we demonstrate that the constituent linear terms in the commutator can be included at no extra cost. As an application, we show how to use the product formulas in a digital protocol for counterdiabatic driving, which increases the fidelity for quantum state preparation. We also discuss applications to quantum simulation of one-dimensional fermion chains with nearest- and next-nearest-neighbor hopping terms, and two-dimensional fractional quantum Hall phases.

Backward iterations for solving integral equations with polynomial nonlinearity

The theory of adjoint operators is widely used in solving applied multidimensional problems with Monte Carlo method. Efficient algorithms are constructed using the duality principle for many problems described in linear integral equations of the second kind. On the other hand, important applications of adjoint equations for designing experiments were suggested by G.Marchuk and his colleagues in their respective works. Some results obtained in these fields are also generalized to the case of nonlinear operators. Linearization methods were mostly used for that purpose. Results for Lyapunov-Schmidt nonlinear polynomial equations were obtained in the theory of Monte Carlo methods. However, many interesting questions in this subject area are remained open. New results about dual processes used for solving polynomial equations with Monte Carlo method are presented. In particular, an adjoint Markov process for the branching process and the corresponding unbiased estimate of the functional of the solution to the equation are constructed in a general form. The possibility of constructing an adjoint operator to a nonlinear one is discussed.

A Highly Efficient Computer Method for Solving Polynomial Equations Appearing in Engineering Problems

A highly efficient two-step simultaneous iterative computer method is established here for solving polynomial equations. A suitable special type of correction helps us to achieve a very high computational efficiency as compared to the existing methods so far in the literature. Analysis of simultaneous scheme proves that its convergence order is 14. Residual graphs are also provided to demonstrate the efficiency and performance of the newly constructed simultaneous computer method in comparison with the methods given in the literature. In the end, some engineering problems and some higher degree complex polynomials are solved numerically to validate its numerical performance.

On highly efficient derivative-free family of numerical methods for solving polynomial equation simultaneously

A highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous iterative method possesses 12th-order convergence locally. Numerical examples and computational cost are given to demonstrate the capability of the method presented.

Computer visualization and dynamic study of new families of root-solvers

The considered topic, the dynamic study of new root solvers using computer tools, is actually a “bridge” between computer science and applied mathematics. The goal of this paper is the construction and dynamic study of two new one-parameter families for solving nonlinear equations using advanced computer tools such as symbolic computation, computer graphics and multi-precision arithmetic. First family, based on Popovski’s third order method (Popovski, 1980), is modified and adapted for finding multiple zeros of differentiable functions. Its dynamic study is performed using basins of attraction and associated quantitative data. In the second part this one-point family for a single zero serves for the derivation of a very efficient family of iterative methods with corrections for the simultaneous determination of all multiple zeros of algebraic polynomials. The convergence analysis, performed by the help of symbolic computation in computer algebra system Mathematica, has shown that the order of convergence of the proposed family is four, five and six, depending of the type of used corrective approximations. A very fast convergence rate is obtained without any additional evaluations of a given polynomial P and its derivatives P′ and P′′, which points to the high computational efficiency of new methods. Choosing different values of the involved parameter, the presented family generates a variety of simultaneous methods. Employing multi-precision arithmetic, it has been shown by numerical experiments that four particular methods from the family produce approximations of very high accuracy. Computer visualization, carried out by plotting trajectories of zero approximations, points to the stability and robustness of the proposed simultaneous methods and indicates a conjecture on their globally convergent properties, one of the most important features of simultaneous methods for solving polynomial equations.

Attasi nD Systems and Polynomial System Solving

Firstly it will be shown that, using the concept of monomial orderings, the classical 1D theory for linear systems generalizes in a very natural way to (autonomous) Attasi nD-systems, giving the Attasi-Hankel matrix, the Attasi transfer function and Attasi state-space realizations. Secondly we will explain how one can associate an Attasi system to any set of polynomial equations having a finite number of solutions and how Attasi realization theory can be used to (1) describe a commutative matrix solution to the set of polynomial equations (this generalizes the Cayley-Hamilton theorem) and (2) find the (scalar) solutions to the set of polynomial equations by computing the joint eigenvalues and eigenvectors of the commuting matrices. Some remarks will be made about how the (discrete time) Attasi system equations can be solved recursively and how that can be used in large-scale eigensolvers. Using the associated Attasi system to a set of polynomial equations also helps to understand various different approaches to polynomial equation solving and multivariate polynomial and rational function minimization.

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Multi-Variant LRP-Polynomials and Homotopy of Polynomial Equations Systems

Particular case of nonlinear equations are polynomial equations, solving algorithms of which are justified and investigated in the most detail. However, a general approach to solving such equations and their systems that could be considered universal for solving most practical problems has not yet been developed. This is an incentive to search for new algorithms, adapted, at least, to solve typical applied problems. This paper is devoted to the development the method of Laguerre's type for solving polynomial equations systems with real coefficients. It is shown that this method is close in form to the method of homotopy, which effective, for example, in solving optimization problems of nonconvex functions. Its efficiency and advantages in comparison with the known methods are demonstrated on examples of the study of mathematical models of real objects and processes.

Automorphisms of Finite Quasi-Groups without Sub-Quasi-Groups

Finite quasi-groups without sub-quasi-groups are considered. It is shown that polynomially complete quasi-groups with this property are quasi-primal. The case in which the automorphism groups act transitively on these quasi-groups is considered. Quasi-groups of prime-power order defined on an arithmetic vector space over a finite field are also studied. Necessary conditions for a multiplication in this space given in coordinate form to determine a quasi-group are found. The case of a vector space over the two-element field is considered in more detail. A criterion for a multiplication given in coordinate form by Boolean functions to determine a quasi-group is obtained. Under certain assumptions, quasi-groups of order 4 determined by Boolean functions are described up to isotopy. Polynomially complete quasi-groups are important in that the problem of solving polynomial equations is NP-complete in such quasi-groups. This property suggests using them for protecting information, because cryptographic transformations are based on quasi-group operations. In this context, an important role is played by quasi-groups containing no sub-quasi-groups.

Solution to Polynomial Equations, a New Approach

A new approach for solving polynomial equations is presented in this study. Two techniques for solving quartic equations are described that are based on a new method which was recently developed for solving cubic equations. Higher order polynomial equations are solved by using a new and efficient algorithmic technique. The proposed methods rely on initially identifying the vicinities of the roots and do not require the use of complicated formulas, roots of complex numbers, or application of graphs. It is proposed that under the stated conditions, the methods presented provide efficient techniques to find the roots of polynomial equations.